Spiralling branes, affine qq-characters and elliptic integrable systems

TL;DR

This paper extends the spiralling branes technique to many-body integrable systems, providing new R-matrix descriptions, exploring elliptic deformations, and establishing noncommutative identities for affine qq-characters.

Contribution

It introduces novel R-matrix formulations for RS Hamiltonians, explores elliptic deformations, and proves noncommutative Jacobi identities for affine qq-characters.

Findings

New R-matrix description of trigonometric RS Hamiltonians

Identification of two elliptic deformations leading to different Hamiltonians

Proof of noncommutative Jacobi identities for affine qq-characters

Abstract

We apply the spiralling branes technique introduced in arXiv:2312.16990 to many-body integrable systems. We start by giving a new R-matrix description of the trigonometric Ruijsenaars-Schneider (RS) Hamiltonians and eigenfunctions using the intertwiners of quantum toroidal algebra. We then consider elliptic deformations of the RS system, elucidate how Shiraishi functions appear naturally in the process and relate them to certain special infinite system of intertwiners of the algebra. We further show that there are two distinguished elliptic deformations, one of which leads to the conventional elliptic RS Hamiltonians, while the other produces trigonometric Koroteev-Shakirov Hamiltonians. Along the way we prove the fully noncommutative version of the "noncommutative Jacobi identities" for affine qq-characters recently introduced by Grekov and Nekrasov.